Multiplying Decimals
Learning Outcomes
- Multiply two decimals together
- Multiply a decimal by 10, 100, or 1000
Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions. Do you remember how to multiply fractions? To multiply fractions, you multiply the numerators and then multiply the denominators. So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side below. Look for a pattern.
A | B | |
---|---|---|
[latex]\left(0.3\right)\left(0.7\right)[/latex] | [latex]\left(0.2\right)\left(0.46\right)[/latex] | |
Convert to fractions. | [latex]\left(\frac{3}{10}\right)\left(\frac{7}{10}\right)[/latex] | [latex]\left(\frac{2}{10}\right)\left(\frac{46}{100}\right)[/latex] |
Multiply. | [latex]\frac{21}{100}[/latex] | [latex]\frac{92}{1000}[/latex] |
Convert back to decimals. | [latex]0.21[/latex] | [latex]0.092[/latex] |
Multiplying Two Numbers
When multiplying two numbers,- if their signs are the same, the product is positive.
- if their signs are different, the product is negative.
Multiply decimal numbers.
- Determine the sign of the product.
- Write the numbers in vertical format, lining up the numbers on the right.
- Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
- Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.
- Write the product with the appropriate sign.
example
Multiply: [latex]\left(3.9\right)\left(4.075\right)[/latex]. Solution[latex]\left(3.9\right)\left(4.075\right)[/latex] | |
Determine the sign of the product. The signs are the same. | The product will be positive. |
Write the numbers in vertical format, lining up the numbers on the right. | |
Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points. | |
Place the decimal point. Add the number of decimal places in the factors [latex]\left(1+3\right)[/latex]. Place the decimal point 4 places from the right. | |
The product is positive. | [latex]\left(3.9\right)\left(4.075\right)=15.8925[/latex] |
try it
[ohm_question]146594[/ohm_question] [ohm_question]146596[/ohm_question]example
Multiply: [latex]\left(-8.2\right)\text{(}5.19\text{).}[/latex]Answer: Solution
[latex]\left(-8.2\right)\left(5.19\right)[/latex] | |
The signs are different. | The product will be negative. |
Write in vertical format, lining up the numbers on the right. | [latex]\begin{array}{c}\hfill 5.19\\ \hfill \underset{\text{_____}}{\times 8.2}\end{array}[/latex] |
Multiply. | [latex]\begin{array}{c}\hfill 5.19\\ \hfill \underset{\text{_____}}{\times 8.2}\\ \hfill 1038\\ \underset{\text{_____}}{4152}\\ \hfill 42558\end{array}[/latex] |
[latex]\begin{array}{c}\hfill 5.19\\ \hfill \underset{\text{_____}}{\times 8.2}\\ \hfill 1038\\ \underset{\text{_____}}{4152}\\ \hfill 42.558\end{array}[/latex] | |
The product is negative. | [latex]\left(-8.2\right)\left(5.19\right)=-42.558[/latex] |
try it
[ohm_question]146597[/ohm_question]example
Multiply: [latex]\left(0.03\right)\text{(}0.045\text{).}[/latex]Answer: Solution
[latex]\left(0.03\right)\left(0.045\right)[/latex] | |
The product is positive. | |
Write in vertical format, lining up the numbers on the right. | |
Multiply. | |
Add zeros as needed to get the [latex]5[/latex] places. | |
The product is positive. | [latex]\left(0.03\right)\left(0.045\right)=0.00135[/latex] |
try it
[ohm_question]146598[/ohm_question]Multiply by Powers of [latex]10[/latex]
In many fields, especially in the sciences, it is common to multiply decimals by powers of [latex]10[/latex]. Let’s see what happens when we multiply [latex]1.9436[/latex] by some powers of [latex]10[/latex]. Look at the results without the final zeros. Do you notice a pattern? [latex-display]\begin{array}{ccc}1.9436\left(10\right)\hfill & =& 19.436\hfill \\ 1.9436\left(100\right)\hfill & =& 194.36\hfill \\ 1.9436\left(1000\right)\hfill & =& 1943.6\hfill \end{array}[/latex-display] The number of places that the decimal point moved is the same as the number of zeros in the power of ten. The table below summarizes the results.Multiply by | Number of zeros | Number of places decimal point moves |
---|---|---|
[latex]10[/latex] | [latex]1[/latex] | [latex]1[/latex] place to the right |
[latex]100[/latex] | [latex]2[/latex] | [latex]2[/latex] places to the right |
[latex]1,000[/latex] | [latex]3[/latex] | [latex]3[/latex] places to the right |
[latex]10,000[/latex] | [latex]4[/latex] | [latex]4[/latex] places to the right |
Multiply a decimal by a power of [latex]10[/latex]
- Move the decimal point to the right the same number of places as the number of zeros in the power of [latex]10[/latex].
- Write zeros at the end of the number as placeholders if needed.
example
Multiply [latex]5.63[/latex] by factors of ⓐ [latex]10[/latex] ⓑ [latex]100[/latex] ⓒ [latex]1000[/latex].Answer: Solution By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.
ⓐ | |
[latex]56.3\left(10\right)[/latex] | |
There is [latex]1[/latex] zero in [latex]10[/latex], so move the decimal point [latex]1[/latex] place to the right. | |
[latex]56.3[/latex] |
ⓑ | |
[latex]5.63\left(100\right)[/latex] | |
There are [latex]2[/latex] zeros in [latex]100[/latex], so move the decimal point [latex]2[/latex] places to the right. | |
[latex]563[/latex] |
ⓒ | |
[latex]5.63\left(1000\right)[/latex] | |
There are [latex]3[/latex] zeros in [latex]1000[/latex], so move the decimal point [latex]3[/latex] places to the right. | |
A zero must be added at the end. | [latex]5,630[/latex] |
Key Takeaways
[ohm_question]146599[/ohm_question]Licenses & Attributions
CC licensed content, Original
- Multiply Decimals by 10, 100, and 1000. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Question ID 146599 , 146598, 146597, 146596,146594. Authored by: Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- License: CC BY: Attribution.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].